Change-point time series specifications constitute flexible models that capture unknown structural changes by allowing for switches in the model parameters. Nevertheless most models suffer from an over-parametrization issue since typically only one latent state variable drives the breaks in all parameters. This implies that all parameters have to change when a break happens. We introduce sparse change-point processes, a new approach for detecting which parameters change over time. We propose shrinkage prior distributions allowing to control model parsimony by limiting the number of parameters which evolve from one structural break to another. We also give clear rules with respect to the choice of the hyper parameters of the new prior distributions. Well-known applications are revisited to emphasize that many popular breaks are, in fact, due to a change in only a subset of the model parameters. It also turns out that sizeable forecasting improvements are made over recent change-point models.